Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{q^2 + 5q - 6}{q^2 - 9q + 8}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 + 5q - 6}{q^2 - 9q + 8} = \dfrac{(q + 6)(q - 1)}{(q - 8)(q - 1)} $ Notice that the term $(q - 1)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q - 1)$ gives: $x = \dfrac{q + 6}{q - 8}$ Since we divided by $(q - 1)$, $q \neq 1$. $x = \dfrac{q + 6}{q - 8}; \space q \neq 1$